There has been great interest in developing linkage analysis methods that allow for the adjustment of covariates, because this type of analysis can potentially allow us greater power to detect genetic effects after adjusting traits for the possible effect of covariates. The object of this study is to compare the performance of two genetic model-free linkage analysis software packages: LODPAL and GENEFINDER. Below we present some brief theoretical background on the two statistical methods implemented in LODPAL and GENEFINDER.

### LODPAL

LODPAL is an affected-relative-pair analysis method using a conditional-logistic model that allows covariates to adjust the relative risks associated with sharing alleles identity by descent (IBD) [1]. Goddard et al. [2] modified the two-parameter method originally described by Olson [1] by assuming a mathematical relationship between the two model parameters λ_{1} and λ_{2}, where λ_{1} is the relative risk for a pair of relatives that shares exactly one allele IBD and λ_{2} is the relative risk for a pair of relatives that shares two alleles IBD. Olson's original method requires two additional parameters for each covariate, while the new method needs only one parameter by using the relationship λ_{2} = 3.634 × λ_{1} - 2.634. This idea of parameter reduction was based on the work described by Whittmore and Tu [3], in which they showed that a minimum-maximum one-parameter ASP LOD score had better power for most genetic models than traditional two-parameter models when assuming a genetic model "approximately half way between a recessive and a dominant mode of inheritance".

### GENEFINDER

Liang et al. [5] developed a multipoint linkage mapping approach for estimating the location of a trait locus using affected sibling pairs. This method makes an assumption that there is no more than one trait locus in the chromosomal region. It has been implemented in the software called GENEFINDER. The primary statistics are the number of alleles shared IBD from multiple markers. The model can be expressed as

E (*S*(*t*) | Φ) = 1 + (1-2θ_{t,τ})^{2}(E(*S*(τ) | Φ) - 1)

= 1 + (1-2θ_{t,τ})^{2} × *C*,

where *S(t)* is the number of alleles shared IBD at an arbitrary locus *t* in the chromosomal region, Φ is the event of affected siblings, θ is the recombination fraction between locus *t* and the unobserved trait locus τ, and *C* is defined as (E(*S*(τ) | Φ) - 1), which is the effect of the unobserved trait locus as characterized by the excessive IBD sharing due to the linkage to an unobserved trait locus. Using the generalized estimating equation (GEE) procedure, we can estimate the parameters of interest, τ and *C*, and their confidence intervals, directly. An interesting feature of this approach is that one can test the null hypothesis of no linkage to this region by testing *C* = 0, which follows a χ^{2} distribution with 1 df. Furthermore, this GEE approach has been extended to incorporate the linkage evidence from unlinked regions [6] and to incorporate covariate information [7]. When incorporating covariate data, the model can be expressed as

E(*S(t)* | x ∈ l,Φ) = 1 + (1 -2θ_{t,τ})^{2}(E(*S*(τ) | x ∈ *l*, Φ) - 1)

= 1 + (1 - 2θ_{t,τ})^{2} × *C*_{
l
},

where *x* is the discrete covariate information, *l*(= 0, 1, 2) is the value of this covariate, and *C*_{
l
}is defined as (E(*S*(τ) | Φ) - 1) for the pairs with a covariate coded as 0, 1, and 2, respectively. Similarly, we can estimate τ, *C*_{
l
}, and their confidence intervals. One can test the null hypothesis of no linkage to this region by testing *C*_{0} = *C*_{1} = *C*_{2} = 0, which follows a χ^{2} distribution with 3 df.